The generator matrix

 1  1  1  1  1  1  1  1  X  X  1  1  1  X  X  1 X^2 X^2  0 X^3  X  X
 0 X^3+X^2 X^3 X^2  0 X^3+X^2 X^3 X^2 X^3+X^2 X^2  0 X^3 X^3+X^2  0 X^3 X^2 X^3+X^2 X^2 X^2 X^2 X^3+X^2 X^2

generates a code of length 22 over Z2[X]/(X^4) who´s minimum homogenous weight is 22.

Homogenous weight enumerator: w(x)=1x^0+56x^22+5x^24+2x^28

The gray image is a linear code over GF(2) with n=176, k=6 and d=88.
As d=88 is an upper bound for linear (176,6,2)-codes, this code is optimal over Z2[X]/(X^4) for dimension 6.
This code was found by Heurico 1.16 in -6.48e-008 seconds.